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e Euler's identity is an identity made by Euler using e, Euler's number. It relates fundamental concepts in math together in a clean and logical way.

Euler's identity

eiπ+1=0 e^{i\pi}+1=0
eiπ+1−1=0−1, Thus eiπ=−1 e^{i\pi}+1-1=0-1,\:Thus\:e^{i\pi}=-1

The beauty of this formula is what it actually does. It connects e; the base of the natural log, i; the imaginary 1D basis vector unit, 1; the multiplicative constant, and 0; the addictive constant. Euler's identity can be expanded to the following:

(1+1n)(n⋅−1⋅4(∑k=0∞(−1)k2k+1)) \left(1+\frac{1}{n}\right)^{\left(n\cdot\sqrt{-1}\cdot 4(\sum_{k=0}^{\infty}\frac{\left(-1\right)^{k}}{2k+1})\right)}

Proof